+ **Nuclear binding energy** is the energy it would take to pull an [[Atomic nucleus]] completely apart into free [[Proton]]s and [[Neutron]]s. Run backwards, it is the energy released when those particles snap together. It is the largest energy budget in ordinary matter, and it is paid for out of [[mass]].+ ## The mass that isn't there++ Weigh a helium-4 atom. Separately weigh its ingredients — two [[hydrogen]] atoms and two free neutrons — and add them up. The parts weigh more than the whole.++ | item | mass (u) |+ |---|---|+ | 2 × ¹H atom | 2.015650 |+ | 2 × free neutron | 2.017330 |+ | **sum of the parts** | **4.032980** |+ | actual ⁴He atom | 4.002603 |+ | **missing mass** | **0.030377** |++ Helium-4 is 0.75% lighter than the pieces it is made of. That missing mass is not lost. It left as [[Energy]] when the nucleus formed, at the exchange rate [[Albert Einstein]] wrote down: **E = mc²**. One atomic mass unit is worth 931.494 103 72 MeV, so the missing 0.030377 u is **28.296 MeV** of binding energy.++ Do that arithmetic for every nucleus and you get the whole subject. The recipe, in atomic masses:++ ```+ B = ( Z·m(¹H) + N·m(neutron) − m(atom) ) × 931.494 MeV/u+ ```++ where Z is the proton count and N the neutron count.++ ## Per nucleon is the number that matters++ Total binding energy just grows with size — [[uranium]] has more of it than helium, the way a brick wall has more mortar than a brick. The useful figure is binding energy **per nucleon**, B/A. That says how tightly each particle is held, and it is what decides which reactions release energy.++ | nuclide | A | B/A (keV) | total B (MeV) |+ |---|---|---|---|+ | ²H (deuterium) | 2 | 1 112.283 | 2.225 |+ | ⁶Li | 6 | 5 332.331 | 31.994 |+ | ⁴He | 4 | 7 073.916 | 28.296 |+ | ¹²C | 12 | 7 680.145 | 92.162 |+ | ²³⁸U | 238 | 7 570.126 | 1 801.69 |+ | ²³⁵U | 235 | 7 590.915 | 1 783.87 |+ | ¹⁶O | 16 | 7 976.207 | 127.619 |+ | ²⁰⁸Pb | 208 | 7 867.453 | 1 636.43 |+ | ⁵⁶Fe | 56 | 8 790.356 | 492.26 |+ | ⁵⁸Fe | 58 | 8 792.253 | 509.95 |+ | **⁶²Ni** | 62 | **8 794.556** | 545.26 |++ ⁶²Ni holds the record: 8 794.556 keV per nucleon, the most tightly bound nuclide known. ⁵⁸Fe and ⁵⁶Fe are close behind, separated by a few keV out of nearly nine thousand. Textbooks usually say "iron" because ⁵⁶Fe is by far the commonest of the three — 91.75% of natural iron — but the strict winner is nickel.++ ++ ## Why the curve has a hump++ Two forces fight over the nucleus.++ - The **[[strong force]]** binds each [[nucleon]] to its immediate neighbours only. It saturates: past a couple of nucleon-widths it stops helping. So its contribution grows roughly with the *number* of nucleons.+ - **Electric repulsion** between protons has no such limit. Every proton pushes every other proton, however far apart, so the penalty grows faster than the size of the nucleus.++ Small nuclei are loosely bound because too many of their nucleons sit on the surface with nothing on the other side to hold. Large nuclei are loosely bound because the protons have piled up too much repulsion. In between, around A ≈ 60, the two effects balance and binding is tightest.++ ## Everything downhill releases energy++ The hump is why both directions of nuclear energy work:++ - **[[nuclear fusion]]** — climbing the steep left slope. Joining light nuclei moves them to higher B/A.+ - **[[k_0FctFP2r7rc|nuclear fission]]** — sliding down the gentle right slope. Splitting a heavy nucleus moves both fragments to higher B/A.++ Both end nearer the peak. Nothing releases energy by fusing iron, which is why stellar cores stop there and then collapse.++ ### Worked example: one fission++ Take a common split of ²³⁵U:++ > ²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n++ | nucleus | total binding energy |+ |---|---|+ | ²³⁵U | 1 783.87 MeV |+ | ¹⁴¹Ba | 1 173.98 MeV |+ | ⁹²Kr | 783.17 MeV |+ | **released** (products − fuel) | **173.28 MeV** |++ That is one nucleus. A kilogram of ²³⁵U holds 2.56 × 10²⁴ of them, so fissioning all of it would release about **71 terajoules** — what an 800-megawatt power station delivers in a full day, out of a lump you could hold in one hand.++ ### Worked example: one star++ The [[Sun]] runs the other slope. Its net reaction turns four hydrogen atoms into one helium atom:++ > 4 ¹H → ⁴He + 2 neutrinos + light++ Weigh both sides: 4 × 1.007825 u in, 4.002603 u out, leaving 0.028697 u — **26.73 MeV** per helium atom made. Per unit of fuel that beats fission by about a factor of nine — 6.6 MeV per nucleon consumed against 0.73 — because hydrogen's climb up the curve is far steeper than uranium's slide down it.++ ## What it is not++ - Not **chemical** bond energy. Burning carbon releases a few eV per atom; nuclear reactions release millions of eV per atom. The gap is about a million to one, and it is entirely because the [[strong force]] is stronger than the electric force that holds molecules together.+ - Not the same as **[[radioactive decay]]** energy. A nucleus can be bound and still unstable — it decays because a *rearrangement* is downhill, not because it is unbound.+ - Not where most of a proton's mass comes from. A [[Proton]]'s own mass is mostly confined [[strong force]] energy among its [[Quark]]s, a different accounting from the binding *between* nucleons described here.++ ## Sources and precision++ The masses above come from the 2020 Atomic Mass Evaluation (AME2020), served through the IAEA Live Chart of Nuclides. Modern mass measurements are extraordinarily good: ⁴He's mass is known to four parts in 10¹¹, which is why binding energies quoted to the keV are meaningful rather than decorative.++ See also: [[half-life]], [[gamma ray]], [[Isotope]], [[Nuclear reactor]], [[Nucleosynthesis]].
History of Nuclear binding energy
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